Minimax filtering for sequential aggregation: Application to ensemble forecast of ozone analyses


Corresponding author: V. Mallet, INRIA, Le Chesnay 78 153, France. (


[1] This paper presents a new algorithm for sequential aggregation of an ensemble of forecasts. At any forecasting step, the aggregation consists of (1) computing new weights for the ensemble members represented by different numerical models and (2) forecasting with a weighted linear combination of the ensemble members. We assume that the time evolution of the weights is described by a linear equation with uncertain parameters and apply a minimax filter (and also Kalman filter, for comparison) in order to estimate the vector of weights given “observations”. The “observation” equation for the filter compares the aggregated forecast with the analysis determined in a data assimilation cycle together with its variance. The minimax approach allows one to work with flexible uncertainty description: deterministic bounding sets for uncertain parameters in weight's equation, and error covariance matrices for the “observational” errors. Our key contribution is an uncertainty estimate of the aggregated forecast, for which we introduce an evaluation test. The performance of the method is assessed for the forecast of ground-level ozone daily peaks over Europe, for the year 2001. Compared to forecasts generated by classical data assimilation, the root mean square error is decreased by 16% for prediction of the analyses and by 20% for prediction of the observations.

1 Introduction

[2] In the task of forecasting the atmospheric state, one can nowadays rely on different sources of information: forecasts provided by numerical models, field observations, and error statistics for both observations and model simulations. The availability of several numerical models reflects the fact that different physical parameterizations were derived to describe the same atmospheric phenomena. Also, the mathematical model could be turned into a numerical model by means of various numerical techniques. In addition, the input data can be provided by different sources and are usually uncertain to some extent. This suggests to consider an ensemble of forecasts: an ensemble brings together various sources of information and allows, therefore, to derive a new forecast which performs better than any individual ensemble member. The improved forecast is usually obtained by taking a linear or convex combination of ensemble members with some weights and is therefore called an aggregated forecast. Now, may be formulated as follows: construct weights such that the corresponding aggregated forecast is close in terms of some performance measure to the given reference which is usually represented by observations. The weights of the combination are updated as soon as new observations become available. So the procedure is referred to as sequential aggregation.

[3] One of the main problems of forecasting algorithms based on ensemble aggregation is uncertainty estimation. Given various numerical models and related observations, with different uncertainty descriptions, one needs to combine these descriptions all together and transform them into an estimate of the uncertainty associated with the aggregated forecast. We stress that the weights of the aggregated forecast change when new observations become available; hence they evolve over time. Thus, the uncertainty transformation should be accomplished by the aggregation algorithm along with the evolution of the weights. In other words, the dynamics of the weights drives the aggregated forecast and its uncertainty estimate. Since this dynamics is uncertain, one needs to assume an appropriate uncertainty description for the weights. We stress that technically the error statistics for observations may be incomparable with uncertainty description for the weight's evolution or even for individual ensemble members: for instance, the measurement error may be stochastic, and the error of the numerical models may be deterministic. The minimax filter can handle such case and provide an uncertainty estimation for the weights (hence for the aggregated forecast as well) in the form of a bounding set.

[4] Another important technical issue to consider is the sparsity (in space) of the observation's network. It may result in weights that are optimal only locally, i.e., at observed locations. This problem is solved using “ensemble forecast of analyses” (EFA) [Mallet, 2010]. In EFA, one first uses a data assimilation algorithm to generate an analysis. At a given date and under given assumptions, the analysis is the a posteriori estimate of the atmospheric state that optimally combines (in the least squares sense) observations, simulations, and error statistics. The main idea of EFA is to forecast forthcoming analyses instead of observations. The analyses are the preferred target because they include all a posteriori knowledge on the atmospheric state, they take into account observational errors, and they provide comprehensive information (i.e., concentrations for all pollutants in all model grid cells). The weights of the aggregated forecast are thus adapted to forecast the analyses instead of the observations, and the weights can depend on space (one weight per model and per state component).

[5] In air quality applications, the aggregation of ensemble simulations has been carried out by different strategies. Delle Monache and Stull [2003] relied on the ensemble mean, where all simulations were given the same weight. Analysis of ensemble mean variance was carried out in Potempski and Galmarini [2009] and Solazzo et al. [2012]. Multimodel forecast based on ensemble median was reported in Riccio et al. [2007]. Bias correction techniques were tested in Monteiro et al. [2013]. Nonstationary weighting procedure for ozone forecasting based on a dynamic linear regression was presented in Pagowski et al. [2006]. Our approach also assumes an equation for the dynamics of the aggregation weights and an observation equation. Note that the latter, in fact, allows one to compare observations (or analyses) and aggregated forecasts corresponding to given weights. However, our uncertainty description differs. In the dynamic linear regression, the errors—in the weights equation and observation equation—are Gaussian and the variance of the observational error is unknown. In contrast, the minimax approach assumes that uncertainty description is given in terms of bounding sets: the errors in the weight equation are elements of a prescribed set. Similarly, the variances of the observational errors may not be prescribed, but they must belong to the given set as well. As a result, the algorithm is robust with respect to uncertainty in observation error covariance matrix unlike Kalman filter: it is well known from the control literature (see, for instance, Shen and Deng [1997] and Başar and Bernhard [1995]), that H2 or Kalman state estimators may be sensitive to uncertainty in the statistical error description. In other words, small perturbations in error covariance matrices may lead to significant deviations in estimates and/or error estimates. Such perturbations are often introduced in practice because the covariance parameters are usually estimated from data. Mallet and Sportisse [2006] used plain least squares methods on a large ensemble. Mallet et al. [2009] applied several machine learning algorithms on the same ensemble, especially a version of the ridge regression with discount in time. Machine learning algorithms are robust, adapted to operational forecasting and guarantee good performance in the long term. However, contrary to dynamic linear regression or minimax approach, they do not evaluate the uncertainty associated with their forecasts. Also, we prove that the weights obtained by means of the discounted ridge regression can be generated by our algorithm for a suitable choice of parameters.

[6] The main contribution of this paper is a minimax aggregation algorithm and uncertainty estimate associated with forecasts together with a simple method to check its reliability. Our aggregation method is based upon a minimax state estimation approach [see Bertsekas and Rhodes, 1971; Nakonechny, 1978; Milanese and Tempo, 1985; Milanese and Vicino, 1991; Kurzhanski and Vályi, 1997]. The estimation problem is defined as follows: given a state equation (the model, for the weights), an observation equation, and error descriptions, one needs to estimate the state of the model assuming that model errors belong to a given bounding set, and observational errors are realizations of random variables with given mean and unknown variance—which is in a given bounded set as well. To solve the estimation problem, we construct a worst-case error which selects the worst possible realization of uncertain parameters and leads to the maximal possible estimation error. The minimax estimate is chosen to have the least possible worst-case estimation error. This, in turn, allows us to construct a set of all possible estimates that are compatible with the model, observed data, and uncertainty description. This set represents, in fact, an uncertainty estimate.

[7] In this work, we construct the minimax estimate for aggregation weights in the form of a linear recursive filter. The so-called state equation (or model) defines the weights dynamics. The so-called observation equation actually involves the analysis, which is supposed to be equal to a linear combination of the ensemble of simulations, plus some unknown error. The filter estimates the aggregation weights which are in turn used to compute an aggregated forecast. The filter also provides an uncertainty estimation for the weights, from which the forecast uncertainty can be derived. In addition, we propose a method to check the reliability of the uncertainty estimation. We stress that the bounding set for the initial weight may be unbounded reflecting the fact that the initial weight may be chosen arbitrarily. This, in turn, proves that the algorithm's convergence does not depend upon a choice of the initial weight. In addition, the algorithm is robust to the variation in observation error covariance matrices unlike Kalman filters as it was mentioned above.

[8] The paper is organized as follows. Section 2 introduces a version of the minimax filter dedicated to sequential aggregation. It explains how to compute and assess the uncertainty estimation and discusses the links to Kalman filter and discounted ridge regression. Section 3 introduces the application to air quality, with further explanations on the EFA strategy. It briefly describes the ensemble simulations and the generation of the analyses. It also gives the parameters related to the minimax filter. Section 4 evaluates the forecast performance and uncertainty estimations. We also address the sensitivity of the results.

2 Sequential Aggregation With Nonstationary Weights

2.1 Notation

[9] Let math formula be a model at time instant t∈{0,...,T−1} that carries out the time integration from time t to time t+1. The model state vector math formula (in our case, the vector of all pollutant's concentrations across the grid) is updated with math formula. Let math formuladenote “the true” state vector of the process. At time t, the observation vector is denoted yt. An observation operator Htmaps the model state space into the observation space so that math formula can be compared with yt.

[10] Let math formula denote an ensemble of forecasts generated by given models math formula, math formula, at time t: math formula. The i-th components of the aforementioned vectors are denoted with an additional subscript: math formula, yi,t, and math formula. Let wt denote the vector of weights math formula. ∥v∥ stands for the 2 norm of the vector v.

2.2 Minimax Weights

[11] Our main objective is to forecast the true process state math formula by means of a linear combination math formula of ensemble members math formula with coefficients math formula. Intuitively, each ensemble member math formularepresents a direction in the state space and, thus, we seek a vector math formula in the subspace spanned by math formulagiving the best (in sense of the minimax criterion) approximation of math formula. Noting that math formula is solely defined by the corresponding weights math formula, we will be looking for a vector of weights which is optimal in the minimax sense. We will refer to the optimal weights as minimax weights.

[12] Model for Weights: We introduce the true weights math formula that yield the best approximation to math formulain the 2 norm sense. We assume that the true weights satisfy the following weights equation:

display math(1)

where A is a M×M matrix, B is a M×M matrix, et is a model error, and eis an uncertain initial condition.

[13] If one assumes that esand et(st) are independent normal random variables, then the weights math formulawould represent a realization of a nonstationary Ornstein-Uhlenbeck process with discrete time. We stress that this is very well aligned with the nonstationary nature of the underlying physical process. However, the normal assumption is not adapted to physical processes that are always modeled with bounded error, while normal random variables can take values outside any bounded set with nonzero probability. Thus, we will assume that et is bounded within a given ellipsoid (see below, inequality (4)). In what follows, we will not rely on any hypothesis about the distribution of the model error. This shows that the minimax weights are robust and the weights' model can be adapted to any real-life application (based on any ensemble).

[14] Observation for Weights: We introduce a relation between the observations and the weights. The observation equation for the weights reads as follows:

display math(2)

where yt is the observation vector, ηtis the observational error (random vector), and the m-th column of Et is the forecast math formulaof yt by the m-th model:

display math(3)

[15] Uncertainty Description: In the minimax setting, the errors are assumed to satisfy the following constraints (also called uncertainty description):

display math(4)

with w0 being our initial guess of the weights, E· standing for the expectation, and Q, Qt and Rtbeing symmetric positive definite matrices. Each constraint defines an ellipsoid where e, et, or ηtlie. We stress that the second constraint is actually on the covariance matrix of ηt. The best estimation of Rt is proportional to the variance of ηt, but any matrix Rt satisfying the constraint can be taken here. Once Rtis fixed, it determines the acceptable range for the variance of ηt.

[16] Minimax Approach: The weights math formulaare said to be minimax if

display math(5)

for any weights wt that follow the model (1) and for any vector math formula. In other words, the minimax weights math formula have the minimal worst-case error (expressed by sup overall uncertain parameters satisfying (4)). Also they do not depend on a particular realization of the uncertain model errors or the observation error covariance matrices and are, therefore, robust to these parameters. In addition, all possible realizations of the weights math formula form a set which is centered at the minimax weights math formula. This set allows to construct the uncertainty estimate for the minimax weights as well as for the aggregated forecast.

[17] Forecasting: We assume that the ensemble of forecasts math formula is available at any time step t. The aggregated forecast math formula is produced with the forecast weight vector math formulaas follows:

display math(6)

if Am is the m-th row of A. Note that math formula corresponds to the expected weights at t+1 according to equation (1).

[18] Minimax Weights: The minimax weights and their uncertainty description math formula are initialized with

display math(7)
display math(8)

and they are updated as follows:

display math(9)
display math(10)

The derivation of these formulae is sketched in Appendix A.

[19] Uncertainty Estimation:math formulais the inverse of the minimax gain matrix, and it provides the uncertainty description for math formula. The forecast weights math formulaare associated with uncertainty description

display math(11)

which is the same as math formula(see equation (9)) with Et+1=0 in order to reflect the absence of observations at the forecast time instant t+1. Note that math formula.

[20] Considering (6), we find that the uncertainty on the i-th component of the aggregated forecast math formula is

display math(12)

so that if the error descriptions (4) are correct, then on average, the truth math formula is guaranteed to satisfy

display math(13)

2.2.1 A Posteriori Evaluation of the Uncertainty Estimation

[21] We propose a method to check the quality of the uncertainty γi,t. It plays the same role as the χ2diagnosis in Kalman filtering [e.g., Ménard et al., 2000]. In our case, the uncertainty description is given by bounded sets, see inequalities (4) and (13), in which we will assume uniform distribution. We therefore assume that every single observation y is uniformly distributed around the unknown truth xtrue: math formula, where ϵ is related to the diagonal elements of Rt. In this paper, we assume ϵis equal to the square root of the corresponding diagonal element of Rt. Similarly, as a consequence of (13), we assume that the truth is uniformly distributed around the minimax estimate math formula: math formula.

[22] Then we compute the probability that the observation y falls outside the interval math formula. In case the observation was perfect (y=xtrue), it should fall inside the interval math formulawith probability 1. With observational errors and given the truth xtrue, the probability that yxtrueϵ,xtrue+ϵ falls outside math formula is equal to math formula, where |a,b|=ba (interval width). Considering that math formula, we find that

display math(14)

This expectation can be compared with the actual frequency in which the observations fall outside the predicted interval math formula. They should be similar if the uncertainty description (4) is reliable.

2.2.2 Minimax Versus Kalman Filter

[23] It is possible to make links with Kalman filter. To do so, we would apply the minimax filter with this uncertainty description instead of inequality (4):

display math(15)

assuming ηs is deterministic in this case.

[24] In the Kalman setting, we would interpret Q, Qs, and Rs as covariance matrices for the initial weights error, for the weights model error, and for the observational error, respectively. Under the weights' equation (1) and the observation equation (2), the Kalman filter and the minimax filter compute the same estimate

display math(16)

The minimax gain matrix math formulareads

display math(17)

This update is the same as in the Kalman filter, but in the Kalman filter, math formula is the variance of the error on math formula.

[25] In this paper, we will report some of the results obtained with the Kalman filter.

2.2.3 Minimax versus Ridge Regression

[26] In the previous work [Mallet et al., 2009; Mallet, 2010], the aggregation weights were computed using the discounted ridge regression:

display math(18)

where λ>0 and ψt>0 is a decreasing sequence of some form [Mallet et al., 2007a].

[27] In case uncertainties are described as inequality (15), both the minimax filter and the Kalman filter lead to the same estimate as the ridge regression at t if A=I, B=0, w0=0, Q=λ−1I, and Rs=(1+ψts)−1I.

3 Experiment Setup for Ozone Ensemble Forecasting

3.1 Ensemble Simulations and Generation of Analyses

[28] This section is essentially a summary of section 3.1 from Mallet [2010] since the experiment setup is the same in this work.

[29] We aim at forecasting ground-level ozone at 15:00 UTC on the next day across western Europe. The ensemble simulations were generated within the Polyphemus platform [Mallet et al., 2007b] for the full year 2001. They were generated and analyzed by Garaud and Mallet[2010]. Each simulation was computed by a different 3-D Eulerian chemistry-transport models. By “model,” we refer to a unique description of the phenomena in terms of physical formulation, numerical discretization, and input data. All models share a horizontal resolution of 0.5°. The meteorological data are from the European Centre for Medium-Range Weather Forecasts. The differences between the models lie in the chemical mechanism (which can be Regional Acid Deposition Model 2 (RADM 2), [Stockwell et al., 1990] or Regional Atmospheric Chemistry Model (RACM) [Stockwell et al., 1997]), the computation of the photolysis rates, the parameterization for vertical diffusion (from Louis[1979] or Troen and Mahrt [1986]), the deposition velocities (Wesely [1989] or Zhang et al. [2003]), the evaluation of the cloud attenuation, the number of vertical layers, and so forth. Twelve input fields, such as boundary conditions, emissions or winds, were perturbed with homogeneous and constant perturbations following either normal or log-normal distributions. The ensemble is composed of 20 members in total. Most of these members were randomly generated using the alternatives and perturbations previously mentioned.

[30] One model (the first member) from the ensemble is used to generate the analyses. Of course, the analyses themselves are not inside the ensemble, but the first member relies on data assimilation until 19:00 UTC. Then a forecast for 15:00 UTC the next day is computed by the model starting from the analysis at 19:00 UTC. This procedure is repeated for every day, so as to mimic an operational cycle. The other members do not benefit from data assimilation.

[31] The analyses are generated for ozone in the three first model layers with the optimal interpolation method. This method computes the so-called best linear unbiased estimator (BLUE) and updates the model state with BLUE whenever observations become available. We chose this method after the work by Wu et al. [2008] in which optimal interpolation gave good results compared to other data assimilation methods.

[32] If we assume that the error on the model state math formulabefore assimilation (the so-called background) has variance math formula and the error on observations ythas variance Ut, then BLUE reads

display math(19)

The observational error variance is diagonal: Ut=rIt. Let lhand lv be respectively the horizontal and vertical distances between two grid cells (in the first three layers where the analysis is computed). The background state error covariance between these points is given in the form

display math(20)

where the decorrelation lengths are set to Lh=1° and Lv=150m. After a χ2diagnosis [e.g., Ménard et al., 2000] applied in Mallet [2010], the variances are set to b=190μg2m−6 and r=51μg2m−6. In this case, the standard deviation of the observational error is about 7.1μgm−3 which is supposed to take into account measurement errors and representativeness errors.

[33] The observations from the European Monitoring and Evaluation Programme monitoring network are assimilated. Every hour, there are available observations from about 90 active stations distributed across Europe—see Figure 4 for the locations of the stations.

[34] The analysis error variance is math formula. Inside the optimal interpolation algorithm, we compute the diagonal of St, which is necessary in the application of minimax-based EFA. In this paper, math formula does not depend on time, but Utdepends on time because of the availability of the monitoring stations.

3.2 Ensemble Forecast of Analyses

[35] In this paper, we apply the ensemble forecast of analyses [Mallet, 2010] so that the aggregation is driven by analyses instead of observations. The aggregation weights are computed independently for each component of the state vector. For each state component i, we have an independent weight equation wi,t+1=Awi,t+Bei,t, and the aggregation follows as math formula. The observations yt at time t are replaced with the (scalar) analysis math formulawhich is available at the same time as the observations (with a small delay due to the BLUE computations (19)). The observation operator from equation (2) reads math formula. The “observation” equation (2) is therefore replaced with

display math(21)

where ηi,t has variance Si,t (the i-th diagonal element of St). Consequently, for each state component, we set Rt (which is now a 1×1 matrix in inequality (4)) to Si,t. Note that the uncertainty description for the error ηi,t is solely due to the uncertainty on the analysis math formula. It does not include a representativeness part since any analysis can be represented as a linear combination of the ensemble of forecasts (unless they are all equal to zero).

[36] We apply then the minimax filter, as previously described, independently in every grid cell. We also apply the Kalman filter for comparison. Applying the filters independently in every grid cell allows better performance, because the weights are adapted to the local situation. In theory, it is possible to use the same weights for part of the components or for all the components, though.

[37] Note that we are driven by the analyses, but we take into account the uncertainties on the analyses (with ηi,t). In fact, we try to forecast the true state math formulain every grid cell i, using the ensemble of forecasts and the analysis which can be seen as some imperfect observation of the true state.

3.3 Aggregation Parameters

[38] The analyses, their variances and the models simulations depend on the grid cell. The other parameters of the aggregation algorithm do not depend on the grid cell. Efficient grid-independent parameters were found after trials and tests in all grid cells at once. The parameters were essentially selected so that the diagnosis (14) is nearly satisfied. They were also tuned to minimize the errors with respect to the analysis and the observations. The robustness of these parameters was evaluated afterward (see section 4.3).

[39] The model for the weights is trivial: A=I(identity matrix) and B=0. In the initial weight vector w0, all components are zero except the component that corresponds to the model that benefits from data assimilation. This model, whose forecasts originate from analyses at 19:00 UTC, is given a weight 1. The first aggregated forecast therefore coincides with the forecast of this model. The value of Q (see equation (4)) is 0.22Iand Qt is set to 0.0152I. The value for Qtwas selected based on the performance of the filter, but the filter shows moderate sensitivity to this parameter—see section 4.3. We stress that the minimax estimate is not sensitive to the initial weights as it is shown by Figure 2 and discussed in section 4.

4 Results

4.1 Forecasting Performance

[40] The forecast performance is evaluated from 1 February 2001 (at 15:00 UTC) to the end of the year (actually, 30 December at 15:00 UTC), hence over 333 days. The month of January serves as a spin-up period for the minimax filter. During the evaluation period, we first compare the individual models from the ensemble with the analyses. The root mean square error (RMSE) is computed with the ground-level ozone concentrations from all n=3082 grid cells, so that the RMSE of the m-th model is math formulaif t=1 corresponds to 1 January 2001 at 15:00 UTC. The RMSE varies greatly among the models of the ensemble: the highest RMSE is 51.2 μgm−3, and the lowest RMSE among the models without assimilation is 16.4 μgm−3. The best model is the first model as it benefits from data assimilation until 19:00 UTC in the previous day. Its RMSE is 13.5 μgm−3. Another reference RMSE is that of the ensemble mean: 18.4 μgm−3. With the parameters from section 3.3, the minimax aggregated forecast has a RMSE of 11.3 μgm−3. The Kalman filter performs equally well (11.3 μgm−3too). Note that this is the same as the performance of the discounted ridge regression, whose RMSE is 11.3 μgm−3 as well.

[41] In Figure 1, we compare the temporal mean of the analyses with (1) the aggregated forecasts, (2) the first model (hence with data assimilation), and (3) the first model without data assimilation. The latter is not part of the ensemble, but it is plotted to illustrate the difference between the model's simulation with and without data assimilation. On average, the minimax forecasts capture all patterns from the analyses, with the right amplitude, while the reference model, with or without assimilation, fails in different regions, e.g., north to Spain.

Figure 1.

Temporal mean of ozone concentrations at 15:00 UTC in μgm−3 for the (top left) reference model without data assimilation, for the (top right) reference model with data assimilation, for the (bottom left) aggregated forecasts, and for the (bottom right) analyses. This figure can be compared with Figure 4 in Mallet [2010], where discounted ridge regression shows similar performance.

[42] Following the classical evaluation approach, we also compute the RMSE against observations, including all observations from the evaluation period. The RMSE of the best model, i.e., the first model, is 19.8 μgm−3. The RMSE of the ensemble mean is 22.5 μgm−3. The RMSE of the aggregated forecasts is 16.2 μgm−3for the minimax filter and 16.6 μgm−3for the Kalman filter. This is higher than the RMSE of the discounted ridge regression, which is 15.6 μgm−3.

4.2 Uncertainty Estimation

[43] A key contribution of the approach is the estimation of the uncertainty on the weights. Figure 2 shows the time evolution of the weights for the 20 models in a grid cell. The model that benefits from data assimilation receives the largest weight during the whole period. Note that the method is not sensitive to the initial weights. In the figure, we also show the evolution of the weights starting from Q−1=0, i.e., with infinite error on the weights. At the end of the time period, a comparable distribution is reached and the overall performance (not shown in this paper) is essentially the same as with Q=0.22I. The uncertainty range contains the values where the true weights can lie according to the uncertainty description for the model errors and the observational errors. The uncertainty estimation corresponds to the first diagonal elements of math formula. The initial value is math formula. It tends to grow in the first steps as the model error, quantified by math formula, accumulates in the weights. At the same time, the assimilation of the analyses tends to decrease the uncertainty on the weights. After about 50 or 70 steps, the amplitude of the uncertainty on the first weight reaches balanced values between the accumulation of model error and the corrections due to the assimilation.

Figure 2.

Time evolution of the minimax weights math formula in the grid cell (j=18, i=13) where j is the index along the latitude and i along the longitude. The first weight (white line on blue background), associated with the model that benefits from data assimilation, is set to 1 in the first guess, while the other weights are set to zero. The blue background represents the set of admissible weights for the first model, i.e., the range of values where the true weight should lie. This range semi-width is the first diagonal element of math formula. On the top, the results were generated with the default configuration. On the bottom, the error on the initial weights is infinite (Q−1=0).

[44] The uncertainty on the concentration can be derived from the uncertainty on the weights, following equation (13). Figure 3 shows the time evolution of the concentrations, for the ensemble models and for the aggregated forecast. They are compared with the analysis, and the uncertainty estimation is provided with the minimax forecast. The grid cell and the time period were selected so that they are representative of the best model RMSE, of the aggregated-forecast RMSE, and of the amount of analyses falling inside the uncertainty bounds. The uncertainty range from the minimax filter is much narrower than the ensemble envelope. Over the whole domain and the evaluation period, the analysis lies in the uncertainty bounds with frequency 0.751, which is similar to the expected value of 0.757 as computed according to the diagnosis (14).

Figure 3.

Ensemble simulations (left) and minimax aggregated forecast (right), in μgm−3, for one grid cell (j=18, i=35) over about 110 days. The analysis is plotted in red on both figures. On the left, the 20 members of the ensemble are plotted in light blue. The blue background fills the space between the upper and lower envelopes of the ensemble (in black). The white line is the first model, which is the best model as it benefits from data assimilation until 19:00 UTC in the previous day. On the right, the minimax aggregated forecast is in white, and the blue background represents the uncertainty.

[45] The uncertainty depends on the grid cell since the aggregation is carried out independently in all grid cells and the analysis variance depends on space. Indeed, the analysis variance is lower at observed locations and higher far off from the observation network. Figure 4 shows the uncertainty map for the aggregated forecast for 1 May. The map shares common features with classical uncertainty maps, e.g., the high uncertainty along the coasts, especially in the south [see Garaud and Mallet, 2011]. At the same time, the uncertainty is reduced around observed locations, as a result of the observations assimilation. This can be clearly seen in Spain where the blue spots correspond to observation stations. The blue spots can be shifted (from the exact station locations) because of the effect of transport. In Figure 5, we show the same uncertainty map, but for 14 February and along with the standard deviation of the ensemble. The standard deviation is only a rough uncertainty representation since the ensemble is not calibrated for uncertainty estimation. The uncertainty estimated by the filter is much lower on average. The patterns are significantly different for most days. Again, lower uncertainty is often found around observed locations.

Figure 4.

Uncertainty map for 1 May 2001 15:00 UTC, in μgm−3. In every grid cell i, it corresponds to γi,t (see equations (12)). The white crosses correspond to the observation stations.

Figure 5.

Uncertainty maps for 14 February 2001 15:00 UTC, in μgm−3. On the top, it is computed by the filter, i.e., it corresponds to γi,t in every grid cell i. On the bottom, it is the standard deviation of the ensemble.

[46] The temporal mean of the forecasts lower bounds, i.e., the temporal mean of math formula (see equations (12)(13)), and the temporal mean of the forecasts upper bounds, i.e., the mean of math formula, are displayed in Figure 6.

Figure 6.

Temporal mean (over the full time period) in μgm−3 of the lower bound (top) and the upper bound (bottom) of the interval math formula where the true concentration is supposed to lie, as introduced in equations (12)(13).

4.3 Sensitivity to the Parameters

[47] We computed the aggregation performance for different values of the weights model error, i.e., different values for Qt=q2I. Table 1shows that the sensitivity of the RMSEs of minimax filter and Kalman filter with respect to q is rather low. Despite the large variations in q, the performance in terms of errors remains significantly better than the performance of the best model. In terms of uncertainty estimation, the criterion (14) is approximately met with values from q=0.002 to q=0.02. As it was mentioned above, the weights are robust to the choice of the initial weight. In other words, the weights model and the observations contain enough information for the weights to converge to the optimal (in the minimax sense) weights over time.

Table 1. Performance of the Aggregation Against the Weights' Model Uncertainty q So That Qt=q2Ia
  1. a

    The best model in the ensemble, which benefits from data assimilation, has a RMSE of 13.5 μgm−3 against analyses and of 19.8 μgm−3 against the observations. The interval math formula is defined in section 2.2, see equations (12) and (13). The lines “Analysis outside math formula” report with what frequency (1) the analyses are supposed to fall outside math formula (line “target”) according to the diagnosis (14) from section 2.2.1, and (2) the analyses are actually found outside math formula in the experiments (line “actual”). For a reliable uncertainty estimation, both values should coincide.

RMSE (analysis), μgm−311.411.011.111.311.612.112.4
RMSE (observations), μgm−316.415.815.916.216.416.817.1
Analysis outside math formula (target)70.9%59.7%31.4%24.3%19.6%13.6%9.7%
Analysis outside math formula (actual)66.0%55.9%33.1%24.9%18.7%9.7%4.1%
RMSE (analysis), μgm−311.511.011.011.311.511.812.1
RMSE (observations), μgm−317.016.316.416.616.917.217.4

5 Conclusions

[48] We introduced a scheme for ensemble aggregation based on minimax filtering. The ensemble of forecasts is linearly combined with dynamical weights evolving over time in order to better forecast the true state. We assumed a linear uncertain model for the aggregation weights and applied a minimax filter on these weights to reduce the uncertainty. The filter computes the set of weights that are compatible with (1) the weight dynamics, (2) the weights' model error that is supposed to belong to a prescribed ellipsoid, and (3) the observations and their errors which also belong to prescribed ellipsoids. Using a similar approach, a Kalman filter can be applied as well, but considering Gaussian errors instead of bounded errors.

[49] A key point of the method is that it produces, along with an aggregated forecast, an estimation of the uncertainty on the weights and therefore of the uncertainty on the forecast concentrations. This uncertainty is described in terms of an ellipsoid where the true weights or concentrations are supposed to lie. We introduced a checking criterion based on the frequency with which the analysis falls outside the ellipsoid for the concentrations. The minimax uncertainty description assumes that all model errors (plus the error on the initial weights) fill the same ellipsoid: in other words, we assume one global bounding set. A more natural uncertainty description might rely on separate ellipsoids (for the model error) on every single time step, but some investigation is required so that the filter does not provide a strong overestimation of the uncertainty.

[50] The method can be applied to forecast at individual observation stations, but in this paper, it is applied in the context of ensemble forecasting of analyses for ground-level ozone over Europe and during a full year. The performance of the filter is similar to that of the learning methods. Contrary to the previous work with learning methods, we have no comparison against all linear combinations constant over the full simulation period, but the filtering approach provides an uncertainty estimation. Furthermore, using appropriate parameters (A=I, B=0), the filter can generate a constant weight vector which can be considered as a mean weight vector. One could think of using this mean weight, for instance, setting A=aI and B=I, and introducing nontrivial systematic model error equal to the mean weight vector.

[51] The analyses lie most of the time in the uncertainty range, and the uncertainty is much lower than the ensemble spread, especially around observed locations. Considering the performance and the availability of an uncertainty estimation, the method could be applied operationally just like ensemble forecast of analyses with machine learning is applied on the Prev'air operational forecasting platform operated by Institut National de l'Environnement Industriel et des Risques.

[52] While the spatial patterns of the analyses are almost perfectly reproduced on average, the day-to-day forecasts of the analyses can show erroneous patterns. Future work should address the aggregation of spatial fields, which should involve weights depending on nonlocal analysis values. Since uncertainty estimation is available, additional information is provided and this may help future work on the forecast of threshold exceedance based on an ensemble. However, the final uncertainty estimations can be accurate only if the analyses error variances are accurate. In this paper, the estimations of the analyses error variances relied on a χ2diagnosis, which is a clear limitation.

Appendix A: Construction of the Minimax Weights

[53] We assume that the true weights math formulaare given in the following form:

display math(A1)

where the vectors e, et, and ηt play the role of uncertain parameters which vary within the prescribed bounds. For simplicity, we derive the filter with w0=0.

[54] At some time t∈{0,...,T}, we will be looking for the estimate of the projection of math formulaonto some , i.e., math formula, in the class of linear functions u defined on observations:

display math

[55] The minimax estimate math formula minimizes the so-called worst-case error:

display math

In what follows, we sketch one way of constructing the minimax estimate. We refer the reader to Zhuk[2010] and Mallet and Zhuk [2010] for the details. Using Cauchy inequality for inner products weighted by Q, Qs, and Rs, one can compute

display math(A2)

where ztis a solution of the so-called adjoint equation

display math(A3)

Now, we see that the minimax estimate math formula solves the so-called linear quadratic optimal control problem that is to minimize quadratic cost function (A2) (which is the worst-case error assigned to the estimate u) over the adjoint equation (A3). Solution to this problem has the following form (see Åström [2006] for details): us=RsEsps where

display math(A4)

After simple algebra, we derive the following representation for the minimax estimate math formula and minimax error math formula:

display math

with math formula and math formula are defined in (7)(10).


[56] The third author was partially supported by ERCIM “Alain Bensoussan” Fellowship Programme, which is supported by the Marie Curie cofunding by the European Commission.